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MAX-MIN DEPENDENCE COEFFICIENTS 15
2. Max-min dependence coefficients
Let X = (X1, . . . , Xd) be a vector of unit Fr´echet random variables, that is, with marginal distribution function F(x) = exp( x 1), x > 0, and G denote the Multivariate Extreme Value distribution of X.
The tail dependence function (Huang, 1992; Schmidt and Stadtmu¨ller, 2006) of G is defined by `(x ,...,x ) = logG(x 1,...,x 1), (x ,...,x ) 2 [0,1)d.
1d1d1d It is a convex function, homogeneous of order one and satisfies
_d j=1
xj `(x1,...,xd)
Xd j=1
xj,
with the lower bound corresponding to X with totally dependent margins and the upper bound to X with independent margins.
The tail dependence function `(x1,...,xd) gives us informaWtion about the probability of occurrence of extreme events for the maximum dj=1 F(Xj). In fact, we have (Schmidt and Stadtmu¨ller, 2006; Ferreira and Ferreira, 2012a, 2012b)
`(x1,...,xd)= lim ⇣ tlogP⇣F(X1)1 x1,...,F(Xd)1 xd⌘⌘ t!1⇣ xt x⌘t
= lim tP F(X1)>1 1 _···_F(Xd)>1 d . t!1 t t
It holds that `(x,...,x) = x`(1,...,1) and, for i(S) = 1 if i 2 S and i(S) = 0 i f i 2/ S ,
`( 1(S),..., d(S)) = ✏(XS), (2.1)
where ✏(XS) is the extremal coe cient of the subvector XS of X with indices
in S (Tiago de Oliveira, 1962-63; Smith, 1990). It takes values in the interval
[1,|S|], where |S| is the number of elementsVin S, with ✏(XS) = 1 when XS has
the minimum copula CXS (u1,...,ud)S = Q uj and ✏(XS) = |S| when XS j2S
has the product copula CXS (u1,...,ud)S = j2S uj. W
Let I = {I1,...,Ip} be a partition of D = {1,...,d}, M(Ij) = i2Ij Xi and
XIj the subvector of X with indices in Ij.
For given j 2 (0,1), j = 1,...,p, we will summarize the extremal de-
pendence among the weighted subvectors 1 XIj , j = 1, . . . , p, through the j
coe cient R(X, , I) defined as follows.
Definition 2.1. Let X be a vector of unit Fr´echet random variables and Mul- tivariateExtremeValuedistribution.Foreach =( 1,..., p)2(0,1)p and
